Simplify; express your answer in exponential form. Assume $a\neq 0, y\neq 0$. $\dfrac{{(a^{-3}y^{-4})^{-2}}}{{a^{5}y^{5}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{-3}y^{-4})^{-2} = (a^{-3})^{-2}(y^{-4})^{-2}}$ On the left, we have ${a^{-3}}$ to the exponent ${-2}$ . Now ${-3 \times -2 = 6}$ , so ${(a^{-3})^{-2} = a^{6}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{-3}y^{-4})^{-2}}}{{a^{5}y^{5}}} = \dfrac{{a^{6}y^{8}}}{{a^{5}y^{5}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{6}y^{8}}}{{a^{5}y^{5}}} = \dfrac{{a^{6}}}{{a^{5}}} \cdot \dfrac{{y^{8}}}{{y^{5}}} = a^{{6} - {5}} \cdot y^{{8} - {5}} = ay^{3}$